Harmonic-Counting Measures and Spectral Theory of Lens Spaces

Abstract

In this article, associated with each lattice $T\subseteq \mathbb{Z}^n$ the concept of a harmonic-counting measure $\nu_T$ on a sphere $S^{n-1}$ is introduced and it is applied to determine the asymptotic behavior of the eigenfunctions of the Laplace-Beltrami operator on a lens space. In fact, the asymptotic behavior of the cardinality of the set of independent eigenfunctions associated with the elements of $T$ which lie in a cone is determined when $T$ is the lattice of a lens space.